This paper studies various algorithmic issues in reconstructing a species tree from gene trees under the duplication and the mutation cost model. This is a fundamental problem in computational molecular biology. Our main results are as follows. 1. A linear time algorithm is presented for computing all the losses in duplications associated with the least common ancestor mapping from a gene tree to a species tree. This answers a problem raised recently by Eulenstein, Mirkin, and Vingron [J. Comput. Bio., 5 (1998), pp. 135-148]. 2. The complexity of finding an optimal species tree from gene trees is studied. The problem is proved to be NP-hard for the duplication cost and for the mutation cost. Further, the concept of reconciled trees was introduced by Goodman et al. and formalized by Page for visualizing the relationship between gene and species trees. We show that constructing an optimal reconciled tree for gene trees is also NP-hard. Finally, we consider a general reconstruction problem and show it to be NP-hard even for the well-known nearest neighbor interchange distance. 3. A new and efficiently computable metric is defined based on the duplication cost. We show that the problem of finding an optimal species tree from gene trees is NP-hard under this new metric but it can be approximated within factor 2 in polynomial time. Using this approximation result, we propose a heuristic method for finding a species tree from gene trees with uniquely labeled leaves under the duplication cost. Our experimental tests demonstrate that when the number of species is larger than 15 and gene trees are close to each other, our heuristic method is significantly better than the existing program in Page's GeneTree 1.0 that starts the search from a random tree.

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UV/vis absorption and fluorescence spectra and fluorescence quantum yields and lifetimes of a series of methano-and pyrrolidino-[60]fullerene derivatives in different solvents are studied systematically. The absorption and fluorescence properties of the derivatives with different substituents are somewhat different from those of [60]fullerene, but very similar among themselves, indicating that the low-lying transitions and the photoexcited state processes are dictated by the electronic structures of functionalized [60]fullerene cages. The results also allow an examination of the issue concerning discrepancies between experimentally determined transition probabilities and those calculated in terms of the Strickler-Berg equation for fullerene molecules. In addition, quenchings of the excited singlet states of the [60]fullerene derivatives by electron donor N,Ndiethylaniline (DEA) and the formation of emissive fullerene-DEA exciplexes in solvents of different polarities are investigated.

A systematic study of the photophysical properties and nonlinear absorptive optical limiting responses of the [60]fullerene dimer (C 60 dimer) and poly[60]fullerene (poly-C 60 ) polymer in room-temperature solutions is reported. The results are compared with those of C 60 and representative C 60 derivatives. While the absorption, fluorescence (spectrum, quantum yield, lifetime), and photoinduced electron-transfer properties of the C 60 dimer are somewhat different from those of C 60 , they are qualitatively similar to those of other C 60 derivatives. The triplet-triplet absorption of the C 60 dimer is noticeably weaker than those of C 60 and other C 60 derivatives, corresponding to lower optical limiting responses of the C 60 dimer at 532 nm. However, the photophysical and nonlinear absorptive optical limiting properties of the poly-C 60 polymer are significantly different. The polymer solution shows much weaker fluorescence, barely detectable triplet-state absorption, and marginal optical limiting response. The different excited-state and nonlinear optical properties of the poly-C 60 polymer are explained in terms of new excited singlet-state decay pathways in the polymer that are not available in monomeric and dimeric C 60 molecules. In addition, the issue concerning significant excited-state fullerene cage-cage interactions in the C 60 dimer and poly-C 60 polymer is discussed.

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A systematic spectroscopic study of a series of C60 derivatives with different cage functionalizations is reported. The absorption spectra and absorptivities of the derivatives in solution were measured and compared. By recording the fluorescence spectra using a near-infrared-sensitive emission spectrometer (extending to 1200 nm), fluorescence quantum yields of the derivatives were determined quantitatively. Fluorescence lifetimes of the derivatives were obtained using the time-correlated single photon counting technique. The results show that both fluorescence quantum yields and lifetimes are rather similar for the different classes of C60 derivatives. The nonlinear absorptive properties of the derivatives were evaluated by optical limiting measurements in solution and in polymer film using the second harmonic of a Q-switched Nd:Yag laser at 532 nm. Effects of different fullerene cage functionalizations on the photophysical properties and optical limiting responses of the C60 derivatives are discussed.

[60]Fullerene−styrene copolymers with [60]fullerene contents up to 50% (wt/wt) were prepared in radical-initiated polymerization reactions. Molecular weights of different fractions of the copolymers were determined by gel permeation chromatography measurements using chloroform and DMSO as mobile phases. Structures of the copolymers with different [60]fullerene contents were characterized by use of proton and 13C NMR, FT-IR, and thermal analysis methods. The mechanistic implication of the proposed copolymer structures is discussed.